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Questions tagged [line-bundles]

For questions about line bundles, that is vector bundles of rank $1$, over topological spaces.

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Classification of line bundles over surfaces

I'm currently trying to understand the blow-up process for 4-manifolds. A step in this journey is to understand, topologically, what happens when you pluck out a 4-ball and replace it with $\mathbb{CP}...
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38 views

Line bundles correspoding to a hyperplane

Assume we have a smooth projective variety $X$ over a field and a hyperplane section $H$ on it. For each Weil divisor on $X$ you can construct a line bundle on $X$. For $H$ this line bundle which is ...
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18 views

Tensor product of very ample line bundle with globally generated line bundle is very ample

I think I solved Exercise II 7.5 (d) from Hartshorne's Algebraic Geometry, but I don't know if I used the hypothesis. Those things alway leaves me doubtful if I made a mistake, so I would like to know ...
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24 views

Sections of a quasi-coherent sheaf along the non-vanishing set of a section of a line bundle.

Let $(X,\mathcal O)$ be a quasicompact, quasiseperated scheme, $\mathcal L$ a line bundle on $X$ and $\mathcal F$ any quasi-coherent sheaf on $X$. Let $s \in \Gamma(X,\mathcal L)$ be any global ...
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1answer
62 views

Line bundle from $\mathcal{O}(p)\cong \mathcal{O}(q)$

Let $X$ be a compact Riemann surface. If $\mathcal{O}(p)\cong \mathcal{O}(q)$. How to see there exists a line bundle $L$ and $s_1,s_2$ two sections of $L$ such that $s_1$ vanishes only at $p$ and $s_2$...
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1answer
47 views

What is the Künneth formula for complete varieties?

I'm reading a part of Mumford's Abelian Varieties, and in the Chapter The theorem of the cube: II he claims that some "Künneth formula" tells us that if $L_1$ is a line bundle on a product $X \times ...
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41 views

Global sections of square root line bundle

Let $C$ be a smooth curve in $\mathbb{P}^2$ over field $\mathbb{C}$. Suppose that I have a very ample line bundle $L$ on $C$ of even degree. Then $L$ has $2^{2g}$ square roots in $Pic\ C$. These are ...
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46 views

Picard group of a conic

Suppose $C$ is a smooth conic in $\mathbb{P^2}$, that is, $C$ is a hypersurface of degree two in the projective plane. Assume that the base field is complex numbers. By the genus degree formula, we ...
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84 views

Examples of non trivial vector bundles

Once you see the notion of vector bundle, next thing you want to see are examples of non trivial vector bundles. Here, I want to collect such examples with justification of one or two lines saying ...
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A question about the definition of ample line bundle.

A question about the definition of ample line bundle. On a Noetherian scheme, a line bundle $L$ is said to be ample if for any coherent sheaf $F$, there exists $n_0\in\mathbb N$ such that $F\otimes ...
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28 views

Closed embedding = very ample line bundle

Let $\pi \colon X \to \mathbb{P}^n$ be a closed embedding given via an invertible sheaf $\cal L$ with global sections $s_0, \dots, s_n$. Thus ${\cal L} \cong \pi^* {\cal O}_X$. Why is ${\cal L} \...
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22 views

Bundle of Endomorphism of Line Bundle always trivial

Let $B$ a topologiocal space (I'm not sure if it should be nice enough for the following statement ... eg paracompact). Let $L$ be a line bundle over $B$. My question is how to see that the morphism ...
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22 views

Show that a space of holomorphic sections of a line bundle is isomorphic to the space of meromorphic functions on a Riemann surface.

Let $\Sigma$ be a Riemann surface, $x \in \Sigma$ and let $z: U \rightarrow D \subset \mathbb{C}$ be a local coordinate system centered at $x$. For every $k \in \mathbb{Z}$, a holomorphic line bundle $...
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1answer
61 views

Pushforwad bundle alond a degree 2 map from $P^1$ to $P^1$

Suppose $f:P^1\rightarrow P^1$ is a degree 2 morphism. Let $L$ be a line bundle on $P^1$ (which is equivalent to a $O(n)$), then what is $f_*L$? As for an open set $U$ we can see the preimage is a ...
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68 views

Base point free for $g^1_2$ for hyperelliptic curve

Let $C$ be a curve of genus lager than 1. $C$ is called hyperelliptic if it contains a $g_2^1$ linear system, meaning that $D$ is of degree $2$ with $dim|D|=2$ if $D$ is such a divisor in this linear ...
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1answer
81 views

Divisor on cubic curve $y^2z=x^3-xz^2$

Let $X$ be the nonsingular cubic curve $y^2z=x^3-xz^2$. Let $P_0=(0,1,0)$. Then the line bundle associated to $3P_0$ is $O_X(1)$. I already know that $3P_0$ is produced by cut the curve with $z=0$. ...
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1answer
77 views

Sections of a line bundle can be extended if the complement is of codimension 2?

Let $X$ be a smooth projective variery over $\mathbb C$, and $U$ be an open subset of $X$ such that the complement of $U$ has (complex) codimension $2$. Let $L$ be a line bundle on $X$. Is the ...
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47 views

Are there any examples of non-holomorphic line bundles?

I am working on a complex geometry course. I want to know some examples of non-holomorphic complex line bundles on complex manifolds, i.e. line bundles that do not admit a holomorphic structure. ...
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80 views

Global sections of a line bundle on a hypersurface in complex projective space.

Let $X \subset \mathbb{P}^n $ be a complex hypersurface of degree $d$. I'm trying to calculate the global sections of $\mathcal{O}(r)_{|X}$ for $r \in \mathbb{Z}$. This is what (I think) I have so ...
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Very ample line bundle isomorphic to a restriction of $\mathcal{O}(-1)$ under an embedding?

I know this claim this wrong since a very ample line bundle isomorphic to a restriction of $\mathcal{O}(1)$ under an embedding. But I can still construct an isomorphism below: Let $L\to X$ be a very ...
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50 views

Canonical bundle on $\Bbb P^1$ questions

Let us consider the canonical bundle $K$ on $\Bbb P^1$. Writing an element in $\Bbb P^1$ as $[z_0:z_1]$ we have $U_0$ where $z_0\ne 0$ and $U_1$ where $z_1\ne 0$ as normal. I can write the ...
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1answer
76 views

Show the space of flat connections $(\mathbb T^2)^N / {\mathfrak S}_N$ is a $(N-1)$-complex projective space

Is it true that $$ \mathbb{T}^2/(\mathbb{Z}/2)=\mathbb{CP}^1=S^2? $$ $$ \mathbb{T}^2/(\mathbb{Z}/2)=\mathbb{CP}^1=S^2? $$ I am trying to digest the following statements: $$ M_{\rm flat} =\mathbb E ...
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1answer
81 views

Zeros of a global section on a holomorphic line bundle

I'm studying an introduction to line bundles and I'm struggling with a particular proof. I'm following some notes that present this theorem: Let $ L \rightarrow X $ be a holomorphic line bundle on a ...
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1answer
43 views

Linking the cohomology of a coherent sheaf on a curve with the cohomology of its restrictions to irreducible components

$\newcommand{\H}{\operatorname{H}}\newcommand{\F}{\mathcal{F}}\newcommand{\G}{\mathcal{G}}\newcommand{\O}{\mathcal{O}}\newcommand{\I}{\mathcal{I}}$ Let $X$ be a projective scheme of dimension one over ...
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1answer
78 views

Regular global sections of invertible sheaves

$\newcommand{\L}{\mathcal{L}}$ $\newcommand{\ox}{\mathcal{O}_X}$ Let $X$ be a projective scheme of dimension one over a field $k$ and let $\L$ be an invertible sheaf on $X$. What are sufficient ...
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93 views

Vanishing locus of global section of invertible sheaf has codimension one

Let $X$ be a projective scheme over some field $k$. Let $\mathcal{L}$ be an ample invertible sheaf on $X$ and let $s \in \mathcal{L}(X)$ denote a global section of $\mathcal{L}$. Let $X_s = \{P \in X \...
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1answer
160 views

Showing the Hopf fibration has no global sections

Let's consider a principle $U(1)$-bundle over $S^2$ with the transition function $g_{\infty 0} = z/|z|$ (it is known as the Hopf fibration). There is a simple topological argument showing that this ...
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1answer
187 views

First Chern class for smooth line bundle

For holomorphic line bundle we define its first Chern class by exponential sequence $$0\to \mathbb Z \to \mathcal O \to \mathcal O^* \to 0 $$ and we can similarly define Chern class for smooth line ...
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Very ample line bundles on a scheme relative to itself

I've been working through Hartshorne section II.5, and am currently thinking about very ample line bundles. It's emphasized that very-ampleness of a line bundle is a relative notion, defined in the ...
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55 views

Differentiable complex line bundles are determined by their Chern class in $H^2(X,\mathbb{Z}$)?

The statement: differentiable complex line bundles are determined by their Chern class in $H^2(X,\mathbb{Z}$) is from Wells' book p.105, which I attached below (also the statement of theorem 4.5 is ...
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1answer
56 views

How does a choice of linearisation of a line bundle fix a moment map?

Let $G \times M \to M$ be a weak Hamiltonian action of a Lie group on a Kahler manifold. Suppose we fix a lift/linearisation of the action of $G$ to an ample line bundle $L \to M$. Apparently this ...
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1answer
165 views

Restriction of the exceptional divisor to itself as a line bundle

If we take a complex projective variety $X$ and blow it up at a point, we get an exceptional divisor $E\cong \mathbb{P}^{n-1}$, where $n=dim(X)$. My question basically regards $\mathcal{O}_{\tilde X}...
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51 views

When can you extend a holomorphic section of the Hyperplane bundle defined on a compact connected submanifold?

Let $S$ be a compact connected complex submanifold of $\mathbb{CP}^N$ (i.e. it is a smooth algberaic variety). Let $f: S \longrightarrow \mathcal{O}(1)|_S$ be a holomorphic section of the Hyperplane ...
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1answer
47 views

Why is : $ K_{ \mathbb{P}^{n} } = -(n+1) H = \mathcal{O}_{ \mathbb{P}^{n} } (-n-1) $?

Consider the canonical bundle $ K_{ \mathbb{P}^{n} } = \bigwedge^n T_{\mathbb{P}^{n}}^* $, which is the line bundle whose sections are holomorphic forms of top degree, expressible locally as $ f(z_1 , ...
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Can I comb unoriented hair on a ball?

I know there is no non-vanishing vector field on $S^2$, so I cannot comb the hair on a ball. (I am treating $S^2$ as a manifold without the ambient space $\mathbb R^3$, which amounts to demanding that ...
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0answers
49 views

Differential geometry of line bundles

I am trying to understand concepts of Gerbes and their differential geometry as generalisation of line bundles and their differential geometry using Hitchin’s notes. I am familiar with concepts of ...
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1answer
33 views

Valuations on sections of line bundles

I'm starting to read about valuations. Given a real valuation $v$ on a (projective complex) variety $X$, centered at $x$, and a line bundle $L$, we can define $v(s)$ for $s \in H^0(X,L)$ by ...
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1answer
133 views

Ample line bundle ?

Let $X$ be a smooth complex projective variety embedded in $\Bbb P^n$. Fix $p \in \Bbb P^n$ and let $Y = \cup_{x \in X} L_x$ where $L_x$ is the line passing through $p$ and $x$. Assume that $L_x \...
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1answer
37 views

Vector Bundle contains the Tautological Line Bundle

Let $E$ be complex vector bundle of rank $r$ over a top. space $X$ and $p:\mathbb{P}(E) \to X$ the associated projective bundle of lines in $E$. Why $p^*E$ containes the linebundle $L = \{(l,u)\in \...
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Generic element of a linear system; Bertini's theorem

Bertini's theorem (see Griffiths and Harris, Principles of Algebraic Geometry) states that:The generic element of a linear system is smooth away from the base locus of the system. My questions are: ...
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1answer
70 views

Ideal sheaf of intersection

Let $p_1,p_2,p_3$ be the points in the general position in $\mathbb{P}^2$ and $\mathcal I$ be their ideal sheaf. I want to find locally free resolution of $\mathcal I$. I can write $0\to\mathcal{O}(-2)...
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1answer
117 views

Chern classes and the Jacobian

I am under the impression that a holomorphic line bundle is determined by its first Chern class. I mean this in the sense that $c_1 : H^1(X,\mathcal{O}_X^\ast) \to H^2(X,\mathbb{Z})$ is injective. I ...
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1answer
87 views

Global section $s$ of ample line bundle such that $X_s$ is everywhere dense

Let $X$ be a projective $k$-scheme of pure dimension $n$ where $k$ is a field. Let $L$ be an ample line bundle on $X$. For all $x \in X$ there is some open neighborhood $U \subseteq X$ and an ...
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1answer
149 views

Why does a globally generated invertible sheaf admit a global section not vanishing on any irreducible component?

Let $L$ be an invertible sheaf on the projective scheme $X$ (of pure dimension $n$). Assume that $L$ is globally generated by the global sections $s_1,\ldots,s_r \in L(X)$. How do we show that ...
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1answer
80 views

$P$ is a branching point of the elliptic curve $(E, P)$

In section 19.9.5 of Vakil, he says an elliptic curve $E$ with a point $P$ admits a double cover of $\mathbb{P}^1$ given by the line bundle $\mathscr{O}_E(2P)$. Then he says $P$ is a branch point ...
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76 views

Local computation of the curvature form of a line bundle

I am trying to show the following proposition: Let $M$ be a differential manifold, $L$ be a line bundle, $D$ be a connection on $L$. Locally we can express $D=d+\theta$ for some one from $\theta$ (...
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1answer
142 views

Very ample line bundle on a projective curve

This is an example in Hartshorne. $X$ is the nonsingular cubic curve $y^2z = x^3 - xz^2$ in $\mathbb{P}_k^2$, and $\mathscr{L} = \mathscr{L}(P_0)$, where $P_0$ is the point $(0,1,0)$. He claims that $\...
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1answer
102 views

Understanding line bundles on $\mathbb{P}_k^1$ using transition functions

Vakil defines $\mathcal{O}_{\mathbb{P}_k^1}(n)$ as follows: it is Spec $k[x_1/x_0]$ on $U_0 = D(x_0)$, and Spec $k[x_0/x_1]$ on $U_1 = D(x_1)$, and the transition function from $U_0$ to $U_1$ is ...
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1answer
66 views

Hyperplane line bundle really defined by some hyperplane

I want to prove the following statement: Let $X$ be a smooth hypersurface in $\mathbb P^n$, $D$ be an effective divisor on $X$. If the associated line bundle $\mathcal L(D)$ is equivalent to $\...
0
votes
1answer
104 views

Picard group of a fibration

Assume that $X$ is a projective variety. Let $Pic(X)$ be its Picard group. Let $E$ be a vector bundle over $X$ say of rank $r$ (for example TX). What is the picard group of the total space of $E$? ...